**Decision theory** is a general approach that helps decision makers make intelligent choices. A decision theory problem typically involves the following elements:

**1** **States of nature:** a set of potential future conditions that affects the results of the decision. For instance, the level of demand (high or low) for condominium units will affect profits after the developer chooses to build a large, medium, or small complex. Thus, we have two states of nature—high demand and low demand.

**2** **Alternatives:** several alternative actions for the decision maker to choose from. For example, the real estate developer can choose between building a large, medium, or small condominium complex. Therefore, the developer has three alternatives—large, medium, and small.

**3** **Payoffs:** a payoff for each alternative under each potential state of nature. The payoffs are often summarized in a **payoff table.** For instance, Table 19.1 gives a payoff table for the condominium complex situation. This table gives the profit1 for each alternative under the different states of nature. For example, the payoff table tells us that, if the developer builds a large complex and if demand for units turns out to be high, a profit of $22 million will be realized. However, if the developer builds a large complex and if demand for units turns out to be low, a loss of $11 million will be suffered.

Table 19.1: A Payoff Table for the Condominium Complex Situation

Once the states of nature have been identified, the alternatives have been listed, and the payoffs have been determined, we evaluate the alternatives by using a **decision criterion.** How this is done depends on the **degree of uncertainty** associated with the states of nature. Here there are three possibilities:

**1** **Certainty:** we know for certain which state of nature will actually occur.

**2** **Uncertainty:** we have no information about the likelihoods of the various states of nature.

**3** **Risk:** the likelihood (probability) of each state of nature can be estimated.

Decision making under certainty

In the unlikely event that we know for certain which state of nature will actually occur, we simply choose the alternative that gives the best payoff for that state of nature. For instance, in the condominium complex situation, if we know that demand for units will be high, then the payoff table (see Table 19.1) tells us that the best alternative is to build a large complex and that this choice will yield a profit of $22 million. On the other hand, if we know that demand for units will be low, then the payoff table tells us that the best alternative is to build a small complex and that this choice will yield a profit of $8 million.

Of course, we rarely (if ever) know for certain which state of nature will actually occur. However, analyzing the payoff table in this way often provides insight into the nature of the problem. For instance, examining the payoff table tells us that, if we know that demand for units will be low, then building either a small complex or a medium complex will be far superior to building a large complex (which would yield an $11 million loss).

Decision making under uncertainty

This is the exact opposite of certainty. Here we have no information about how likely the different states of nature are. That is, we have no idea how to assign probabilities to the different states of nature.

In such a case, several approaches are possible; we will discuss two commonly used methods. The first is called the **maximin criterion.**

**Maximin:** Find the worst possible payoff for each alternative, and then choose the alternative that yields the maximum worst possible payoff.

For instance, to apply the maximin criterion to the condominium complex situation, we proceed as follows (see Table 19.1):

**1** If a small complex is built, the worst possible payoff is $8 million.

**2** If a medium complex is built, the worst possible payoff is $5 million.

**3** If a large complex is built, the worst possible payoff is −$11 million.

Since the maximum of these worst possible payoffs is $8 million, the developer should choose to build a small complex.

The maximin criterion is a *pessimistic approach* because it considers the worst possible payoff for each alternative. When an alternative is chosen using the maximin criterion, the actual payoff obtained may be higher than the maximum worst possible payoff. However, using the maximin criterion assures a “guaranteed minimum” payoff.

A second approach is called the **maximax criterion.**

**Maximax:** Find the best possible payoff for each alternative, and then choose the alternative that yields the maximum best possible payoff.

To apply the maximax criterion to the condominium complex situation, we proceed as follows (see Table 19.1):

**1** If a small complex is built, the best possible payoff is $8 million.

**2** If a medium complex is built, the best possible payoff is $15 million.

**3** If a large complex is built, the best possible payoff is $22 million.

Since the maximum of these best possible payoffs is $22 million, the developer should choose to build a large complex.

The maximax criterion is an *optimistic approach* because we always choose the alternative that yields the highest possible payoff. This is a “go for broke” strategy, and the actual payoff obtained may be far less than the highest possible payoff. For example, in the condominium complex situation, if a large complex is built and demand for units turns out to be low, an $11 million loss will be suffered (instead of a $22 million profit).

Decision making under risk

In this case we can estimate the probability of occurrence for each state of nature. Thus, we have a situation in which we have more information about the states of nature than in the case of uncertainty and less information than in the case of certainty. Here a commonly used approach is to use the **expected monetary value criterion.** This involves computing the expected monetary payoff for each alternative and choosing the alternative with the largest expected payoff.

The expected value criterion can be employed by using *prior probabilities.* As an example, suppose that in the condominium complex situation the developer assigns prior probabilities of .7 and .3 to high and low demands, respectively. We find the expected monetary value for each alternative by multiplying the probability of occurrence for each state of nature by the payoff associated with the state of nature and by summing these products. Referring to the payoff table in Table 19.1, the expected monetary values are as follows:

Small complex: Expected value = .3($8 million) + .7($8 million) = $8 million

Medium complex: Expected value = .3($5 million) + .7($15 million) = $12 million

Large complex: Expected value = .3(−$11 million) + .7($22 million) = $12.1 million

Choosing the alternative with the highest expected monetary value, the developer would choose to build a large complex.

Remember that the expected payoff is not necessarily equal to the actual payoff that will be realized. Rather, the expected payoff is the long-run average payoff that would be realized if many identical decisions were made. For instance, the expected monetary payoff of $12.1 million for a large complex is the average payoff that would be obtained if many large condominium complexes were built. Thus, the expected monetary value criterion is best used when many similar decisions will be made.

Using a decision tree

It is often convenient to depict the alternatives, states of nature, payoffs, and probabilities (in the case of risk) in the form of a **decision tree** or **tree diagram.** The diagram is made up of **nodes** and **branches.** We use square nodes to denote decision points and circular nodes to denote chance events. The branches emanating from a decision point represent alternatives, and the branches emanating from a circular node represent the possible states of nature. Figure 19.2 presents a decision tree for the condominium complex situation (in the case of risk as described previously). Notice that the payoffs are shown at the rightmost end of each branch and that the probabilities associated with the various states of nature are given in parentheses corresponding to each branch emanating from a chance node. The expected monetary values for the alternatives are shown below the chance nodes. The double slashes placed through the small complex and medium complex branches indicate that these alternatives would not be chosen (because of their lower expected payoffs) and that the large complex alternative would be selected.

Figure 19.2: A Decision Tree for the Condominium Complex Situation

A decision tree is particularly useful when a problem involves a sequence of decisions. For instance, in the condominium complex situation, if demand turns out to be small, it might be possible to improve payoffs by selling the condominiums at lower prices. Figure 19.3 shows a decision tree in which, after a decision to build a small, medium, or large condominium complex is made, the developer can choose to either keep the same prices or charge lower prices for condominium units. In order to analyze the decision tree, we start with the last (rightmost) decision to be made. For each decision we choose the alternative that gives the highest payoff. For instance, if the developer builds a large complex and demand turns out to be low, the developer should lower prices (as indicated by the double slash through the alternative of same prices). If decisions are followed by chance events, we choose the alternative that gives the highest expected monetary value. For example, again looking at Figure 19.3, we see that a medium complex should now be built because of its highest expected monetary value ($14.1 million). This is indicated by the double slashes drawn through the small and large complex alternatives. Looking at the entire decision tree in Figure 19.3, we see that the developer should build a medium complex and should sell condominium units at lower prices if demand turns out to be low.

Figure 19.3: A Decision Tree with Sequential Decisions

Sometimes it is possible to determine exactly which state of nature will occur in the future. For example, in the condominium complex situation, the level of demand for units might depend on whether a new resort casino is built in the area. While the developer may have prior probabilities concerning whether the casino will be built, it might be feasible to postpone a decision about the size of the condominium complex until a final decision about the resort casino has been made.

If we can find out exactly which state of nature will occur, we say we have obtained **perfect information.** There is usually a cost involved in obtaining this information (if it can be obtained at all). For instance, we might have to acquire an option on the lakefront property on which the condominium complex is to be built in order to postpone a decision about the size of the complex. Or perfect information might be acquired by conducting some sort of research that must be paid for. A question that arises here is whether it is worth the cost to obtain perfect information. We can answer this question by computing the **expected value of perfect information,** which we denote as the **EVPI.** The EVPI is defined as follows:

EVPI = expected payoff under certainty − expected payoff under risk

For instance, if we consider the condominium complex situation depicted in the decision tree of Figure 19.2, we found that the expected payoff under risk is $12.1 million (which is the expected payoff associated with building a large complex). To find the expected payoff under certainty, we find the highest payoff under each state of nature. Referring to Table 19.1, we see that if demand is low, the highest payoff is $8 million (when we build a small complex); we see that if demand is high, the highest payoff is $22 million (when we build a large complex). Since the prior probabilities of high and low demand are, respectively, .7 and .3, the expected payoff under certainty is .7($22 million) + .3($8 million) = $17.8 million. Therefore, the expected value of perfect information is $17.8 million − $12.1 million = $5.7 million. This is the maximum amount of money that the developer should be willing to pay to obtain perfect information. That is, the land option should be purchased if it costs $5.7 million or less. Then, if the casino is not built (and demand is low), a small condominium complex should be built; if the casino is built (and demand is high), a large condominium complex should be built. On the other hand, if the land option costs more than $5.7 million, the developer should choose the alternative having the highest expected payoff (which would mean building a large complex—see Figure 19.2).

Finally, another approach to dealing with risk involves assigning what we call **utilities** to monetary values. These utilities reflect the decision maker’s attitude toward risk: that is, does the decision maker avoid risk or is he or she a risk taker? Here the decision maker chooses the alternative that **maximizes expected utility.** The reader interested in this approach is referred to Section 19.4.

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